\(L^p\)-convergence of greedy algorithm by generalized Walsh system (Q764978)

Let \((\psi_n)_{n=0}^\infty\) be a generalized Walsh system of order \(a\in \mathbb N\), \(a\geq 2\). The author proves two theorems. Theorem 1. Let \(p>2\). For every \(\varepsilon>0\) and \(f\in L^p(0,1)\) there is a function \(g\in L^p(0,1)\) such that \(|\{x\in[0,1]:\;g\neq f\}|<\varepsilon\) and the greedy algorithm of \(g\) converges in \(L^p\). Theorem 2. Let \(p>2\). For any \(\varepsilon>0\) and \(f\in L^p(0,1)\) there is a function \(g\in L^p(0,1)\) such that \(|\{x\in[0,1]:\;g\neq f\}|<\varepsilon\) and such that the sequence \(\{|c_k(g)|, K\in \text{spec}(g)\}\) is monotonically decreasing.